ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5209 on D5407: Positive real number
Subresult of R5210: Tight upper bound to a product of two unsigned real numbers

Of all rectangles with a given perimeter, the square maximizes the area

Formulation 0

Let $a, b, c \in (0, \infty)$ each be a D5407: Positive real number such that

(i)	\begin{equation} 2 a + 2 b = c \end{equation}

Then

(1)	\begin{equation} a b \leq \left( \frac{c}{4} \right)^2 \end{equation}
(2)	\begin{equation} a b = \left( \frac{c}{4} \right)^2 \quad \iff \quad a = b \end{equation}

Proofs

Proof 1

Let $a, b, c \in (0, \infty)$ each be a D5407: Positive real number such that

(i)	\begin{equation} 2 a + 2 b = c \end{equation}

Since \begin{equation} \left( \frac{c}{4} \right)^2 = \left( \frac{2 a + 2 b}{2 \cdot 2} \right)^2 = \left( \frac{a + b}{2} \right)^2 \end{equation} then this result is a special case of R5210: Tight upper bound to a product of two unsigned real numbers. $\square$